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%I #17 Mar 08 2020 00:21:26
%S 1,1,2,1,6,3,1,14,22,4,1,30,105,65,5,1,62,416,581,171,6,1,126,1491,
%T 3920,2695,420,7,1,254,5034,22506,29310,11180,988,8,1,510,16365,
%U 116667,256317,188361,43041,2259,9,1,1022,51892,564667,1945297,2419897,1090135
%N Triangle of numbers related to Eulerian numbers.
%D D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
%H Andrew Howroyd, <a href="/A046803/b046803.txt">Table of n, a(n) for n = 1..1275</a>
%F T(n, k) = Sum_{i=1..n} binomial(n,i) * A008292(n-i, k-1).
%F E.g.f.: exp(x*y)*(exp(x)-1)*(y-1)/(y*exp(x)-exp(x*y)). - _Vladeta Jovovic_, Sep 20 2003
%e Triangle begins
%e 1;
%e 1, 2;
%e 1, 6, 3;
%e 1, 14, 22, 4;
%e 1, 30, 105, 65, 5;
%e 1, 62, 416, 581, 171, 6;
%e 1, 126, 1491, 3920, 2695, 420, 7;
%e ...
%t egf = Exp[x*y]*(Exp[x]-1)*((y-1)/(y*Exp[x] - Exp[x*y])); row[n_] := Last[ CoefficientList[ Series[egf, {x, 0, n}, {y, 0, n}], {x, y}]]*n!; Flatten[ Table[ row[n], {n, 1, 10}]] (* _Jean-François Alcover_, Dec 20 2012, after _Vladeta Jovovic_ *)
%o (PARI) T(n)={my(A=O(x*x^n)); [Vecrev(p) | p<-Vec(serlaplace(exp(x*y + A)*(exp(x + A)-1)*(y-1)/(y*exp(x + A)-exp(x*y + A))))]}
%o { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Mar 07 2020
%o (PARI) \\ here U(n,k) is A008292.
%o U(n, k)={sum(j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))};
%o T(n, k)={sum(i=1, n, binomial(n,i)*U(n-i, k-1))} \\ _Andrew Howroyd_, Mar 07 2020
%Y Row sums give A002627.
%Y Cf. A008292 (Eulerian numbers), A046802.
%K nonn,tabl,easy,nice
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Sep 20 2003
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